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Star systems are notable for being the areas from which the player zooms in to interact with a planet.

The stars themselves serve few purposes, other than housing the various planets of the galaxy and serving as the primary mode of transportation i.e. jumping from star to star to travel the galaxy.
Will Wright in Gaming Steve's interview said about the number of stars and worlds in the Spore galaxy:

"Oh, well, there will be probably well over a hundred thousand. I mean, far more than a player could ever visit. Which is why there is no point to having another galaxy. If there are going to be thousands of stars in a galaxy there is no way that one person in their lifetime could ever visit them all."1

There is no real difference between a moon and a planet. Colonization, T scores, as well as all other aspects of planets, are the same on moons. The only difference is that they appear slightly smaller.

Galactic core is a supermassive black hole in the center of galaxy. To get there is the final purpose of the game. You'll need some time and decent amount of repair kits to survive while fighting/allying with Grox. Additional information is in the Galactic Core article.